Gravitation in terms of observables: breathing new life into a bold proposal of Mandelstam

In the 1960’s Stanley Mandelstam set out to reformulate gravity and gauge theories in terms of observable quantities. The quantities he chose are curves, but specified intrinsically. The simplest way of understanding what does “specified intrinsically” means is to think how the trajectory of a car is specified by a GPS unit. The unit will give commands “turn right”, “advance a certain amount”, “turn left”. In this context “right” and “left” are not with respect to an external coordinate system, but with respect to your car. The list of commands would remain the same whatever external coordinate system one chooses (in the case of a car it could be a road marked in kilometres or miles, for instance). The resulting theories are therefore automatically invariant under coordinate transformations (invariant under diffeomorphisms). They can therefore constitute a point of departure for the quantization of gravity radically different from other ones. For instance, they would share in common with loop quantum gravity that both are loop-based approaches. However, in loop quantum gravity one has to implement the symmetry of the theory under diffeomorphisms. Intrinsically defined loops, on the other hand, are space-time diffeomorphism invariant, therefore such a symmetry is already implemented. It is well known that in loop quantum gravity diffeomorphism invariance is key in selecting in almost unique way the inner product of the theory and therefore on determining the theory’s Hilbert space. Intrinsically defined loops are likely to be endowed with a very different inner product and Hilbert space structure. In fact, since the loops in the Mandelstam approach are space-time ones it lends itself naturally to an algebraic space-time covariant form of quantization.

How a finite loop is generated by the infinitesimal generators.

Mandelstam’s original proposal suffered from a drawback. He was not able to find a way to determine if two intrinsically defined loops ended at the same point. Since in this approach space-time points are derived entities from the intrinsically defined loops, that impediment prevented him from making contact with more traditional approaches where space-time is the central element. In this paper we describe a solution to the problem using the group of loops. The latter was introduced in the context of describing Yang-Mills theories (and later gravity) in terms of holonomies. Holonomies can be composed, have an identity element and an inverse, allowing to define a group structure for curves through their holonomies. This group structure admits infinitesimal generators that amount to adding infinitesimal loops to existing loops. With this structure one can characterise two curves as ending at the same point if they differ by a loop.

With the main obstruction to Mandelstam’s approach eliminated this opens the possibility for a new form of quantizing gravity in which space-time is a derived structure and the central structure are the intrinsically defined loops. Among interesting concepts that arise is that in the quantum theory the notion of space-time point becomes fuzzy. Such fuzziness may have novel consequences that we intend to explore in forthcoming publications.